Holt Geometry

2-4 Biconditional Statements and Definitions2-4 Biconditional Statements

and Definitions

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

2-4 Biconditional Statements and Definitions

Warm UpBe prepared to state a conditional statement

from each of the following.

1. The intersection of two lines is a point.

2. An odd number is one more than a multiple of 2.

3. Write the converse of the conditional “If Pedro lives in Chicago, then he lives in Illinois.” Find its truth value.

If two lines intersect, then they intersect in a point.

If a number is odd, then it is one more than a multiple of 2.

If Pedro lives in Illinois, then he lives in Chicago; False.

Holt Geometry

2-4 Biconditional Statements and Definitions

SWBAT write and analyze biconditional statements.

Objective

HW 2.4 Page 99{1,3,7,9,11,16, 23, 25, 27, 31, 41,43}Don’t forget to use the back of the text to check and score your HW before class.

Holt Geometry

2-4 Biconditional Statements and Definitions

biconditional statementdefinitionpolygontrianglequadrilateral

Vocabulary

Holt Geometry

2-4 Biconditional Statements and Definitions

When you combine a conditional statement and its converse, you create a biconditional statement.

A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.”

Holt Geometry

2-4 Biconditional Statements and Definitions

p q means p q and q p

The biconditional “p if and only if q” can also be written as “p iff q” or p q.

Writing Math

Holt Geometry

2-4 Biconditional Statements and Definitions

Write the conditional statement and converse within the biconditional.

Example 1B: Identifying the Conditionals within a Biconditional Statement

A solution is neutral its pH is 7.

Conditional: If a solution is neutral, then its pH is 7.

Converse: If a solution’s pH is 7, then it is neutral.

Holt Geometry

2-4 Biconditional Statements and Definitions

Check It Out! Example 1a

An angle is acute iff its measure is less than 90°.

Write the conditional statement and converse within the biconditional.

Holt Geometry

2-4 Biconditional Statements and Definitions

Check It Out! Example 1b

Cho is a member if and only if he has paid the $5 dues.

Write the conditional statement and converse within the biconditional.

Conditional: If Cho is a member, then he has paid the $5 dues.

Converse: If Cho has paid the $5 dues, then he is a member.

Holt Geometry

2-4 Biconditional Statements and Definitions

Check It Out! Example 2a

If it is July 4th, then it is Independence Day.

For the conditional, write the converse and a biconditional statement.

Converse: If it is Independence Day, then it is July 4th.

Biconditional: It is July 4th if and only if it is Independence Day.

Holt Geometry

2-4 Biconditional Statements and Definitions

For a biconditional statement to be true, both the conditional statement and its converse must be true.

If either the conditional or the converse is false, then the biconditional statement is false.

Holt Geometry

2-4 Biconditional Statements and Definitions

Determine if the biconditional is true. If false, give a counterexample.

Example 3B: Analyzing the Truth Value of a Biconditional Statement

A person sleeps they are breathing.

Conditional: If a person is sleeping, then they are breathing.

The conditional is true.

Converse: If a person is breathing, then they are sleeping.

The converse is false.

Since the conditional and its converse are not both true, the biconditional is false.

Holt Geometry

2-4 Biconditional Statements and Definitions

Check It Out! Example 3a

An angle is a right angle iff its measure is 90°.

Determine if the biconditional is true. If false, give a counterexample.

Conditional: If an angle is a right angle, then its measure is 90°.

The conditional is true.

Converse: If the measure of an angle is 90°, then it is a right angle.

The converse is true.

Since both are true, the biconditional is true.

Holt Geometry

2-4 Biconditional Statements and Definitions

In geometry, biconditional statements are used to write definitions.

A definition is a statement that describes a mathematical object and can be written as a true biconditional.

Holt Geometry

2-4 Biconditional Statements and Definitions

Think of definitions as being reversible.

Helpful Hint

Holt Geometry

2-4 Biconditional Statements and Definitions

In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments.

Holt Geometry

2-4 Biconditional Statements and Definitions

A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon.

Holt Geometry

2-4 Biconditional Statements and Definitions

Write each definition as a biconditional.

Example 4: Writing Definitions as Biconditional Statements

A. A triangle is a three-sided polygon.

B. A right angle measures 90°.

A figure is a triangle if and only if it is a 3-sided polygon.

An angle is a right angle if and only if it measures 90°.

Holt Geometry

2-4 Biconditional Statements and Definitions

Check It Out! Example 4

4a. A quadrilateral is a four-sided polygon.

4b. The measure of a straight angle is 180°.

Write each definition as a biconditional.

A figure is a quadrilateral if and only if it is a 4-sided polygon.

An is a straight if and only if its measure is 180°.

Holt Geometry

2-4 Biconditional Statements and Definitions

Lesson Quiz

1. For the conditional “If an angle is right, then its measure is 90°,” write the converse and a biconditional statement.

2. Determine if the biconditional “Two angles are complementary if and only if they are both

acute” is true. If false, give a counterexample.False; possible answer: 30° and 40°

Converse: If an measures 90°, then the is right. Biconditional: An is right iff its measure is 90°.

3. Write the definition “An acute triangle is a triangle with three acute angles” as a biconditional.

A triangle is acute iff it has 3 acute s.